Yeah, this all makes sense if the intention is to change the name of the device.
Netflix (the service) has an app named Netflix. You access Netflix via Netflix on... XYZ. Same goes for basically every other streaming service.
So Apple TV the service on Apple TV the app makes perfect sense if you are thinking about accessing their streaming service via other set tops where Apple TV the app is available.
My guess is that the Apple TV set top will be renamed to something else, perhaps "Apple Home".
Then it would be "Access Apple TV via the app on your Apple Home device" and the merging/conflation of "Apple TV subscription via the Apple TV app" will make perfect sense the same way you would say "Access Netflix via the app on your Apple Home device".
My guess is that "tvOS" will be renamed "homeOS" to go with it.
Apple Home is already the name of their app for smart home stuff.
The Netflix comparison doesn’t quite work with Apple TV. I have an Apple TV (the device) and I don’t just use Apple TV (the app) to access Apple TV (the service). Apple TV (the app) is also where I need to go to buy/rent movies from the iTunes Store, watch Apple keynotes, and it can also be a place to aggregate content in a single UI from a bunch of different streaming services (notably not Netflix, they opted out). Apple TV (the service) is just one feature of Apple TV (the app), at least when it’s running on Apple TV (the device).
These distinctions do matter, due to Apple trying to consolidate everything into that single app. They sunset multiple other apps with its release. I actually find the app pretty hard to use as a result.
Netflix (the service) has an app named Netflix. You access Netflix via Netflix on... XYZ. Same goes for basically every other streaming service.
So Apple TV the service on Apple TV the app makes perfect sense if you are thinking about accessing their streaming service via other set tops where Apple TV the app is available.
My guess is that the Apple TV set top will be renamed to something else, perhaps "Apple Home".
Then it would be "Access Apple TV via the app on your Apple Home device" and the merging/conflation of "Apple TV subscription via the Apple TV app" will make perfect sense the same way you would say "Access Netflix via the app on your Apple Home device".
My guess is that "tvOS" will be renamed "homeOS" to go with it.